# Connected sum of two complex projective planes with same orientation

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## Definition

This topological space is defined as the connected sum of two copies of the complex projective plane $\mathbb{P}^2(\mathbb{C})$, where they are glued with the same orientation.

## =Homology groups

The homology groups are as follows:

$Z}, & \qquad p = 0,4 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 2 \\ 0, & \qquad \operatorname{otherwise}\\\end{array$

### Cohomology groups

The cohomology groups are as follows:

$Z}, & \qquad p = 0,4 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 2 \\ 0, & \qquad \operatorname{otherwise}\\\end{array$

The cohomology ring is as follows:

$H^*(M) = \mathbb{Z}[x,y]/(x^2 - y^2, x^3,y^3)$

### Homotopy groups

The fundamental group is the trivial group.